The counter-rotating term exp\((2i\omega _pt)\)  causes rotating terms proportional to exp\([\pm i(\omega _{n}\pm 2\omega _p)t]\) , which is usually a fast rotating term (\(2\omega _p\gg \omega _n\) ) that can be ignored. For example in superconducting circuit (atom) experiments, the frequency of near-resonant microwave (laser) is several GHz (THz) [1]}, [2]}, however, the modulation frequency is about dozens of MHZ [3]}. Another condition for RWA to be applicable in Eq. (REF ) is \(\Omega _{pn}\ll \omega _n+\omega _p\) , which can be simplified to \(\Omega _p\ll \omega _p\approx \omega _{10}\) , and this is the scenario we considered. Note that here we focus on the situation for small n, because the corresponding Rabi frequency \(\Omega _{pn}\)  decreases sharply as n increases. Therefore, in our Floquet system, the RWA is validly and in \(\omega _p\) 's rotating frame, Eq. (REF ) becomes
\(H(t)_{\text{RWA}}= H_0-[\frac{\Omega _{p}(t)}{2}|0\rangle \langle 1|+h.c.].\) 
